From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 26

Box 1.12 (Eckert II, the fourth problem).

(tr[C l G − l ]) 2 =4det[C l G − l ] , Λ 1 = Λ 2 ⇔ (1.88)

det[C l G − l ]=1 ,

(1.89)

(tr[C l G − l ( Λ =0 ,Φ =0)]) 2 = 64 + 9 π 2

2

=4det[C l G − l ]=4 ⇒ Λ 1 = Λ 2 . (1.90)

24 π

Example 1.6 (Orthogonal projection of points of the sphere onto the equatorial plane through
the origin).

Let us assume that we make an orthogonal projection of points of the northern hemisphere onto
the equatorial plane

2 R + .Figures 1.10 and 1.11 illustrate such
an azimuthal projection by means of polar coordinate lines, shorelines, and right Tissot ellipses
of distortion. The mapping equations are given by x = X, y = Y, Z > 0 ,x = R cos Φ cos Λ, y =
R cos Φ sin Λ .

2
O

P through the origin O of the plane S

End of Example.

S 2 R + onto the tangent plane P 2 O at the North Pole, Fig. 1.10. Orthogonal projection of points of the sphere

shorelines, right Tissot ellipses of distorsion

We pose two problems. (i) Derive the right Cauchy-Green deformation tensor. (ii) Solve the right
general eigenvalue-eigenvector problem.